Incomplete vertices in the prime graph on conjugacy class sizes of finite groups.
type: Item
instance of: scholarly article
title: Incomplete vertices in the prime graph on conjugacy class sizes of finite groups.
zbMATH Open document ID: 1282.20027
author: Carlo Casolo
author: Silvio Dolfi
author: Emanuele Pacifici
author: Lucia Sanus
published in: Journal of Algebra
publication date: 2013-08-26T00:00:00Z
full work available at URL: S0021869312005121
reviewed by: Antonio Beltrán
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6201455
zbMATH Keywords: finite groups
zbMATH Keywords: conjugacy class sizes
zbMATH Keywords: prime graphs
zbMATH Keywords: metabelian Hall subgroups
MaRDI profile type: Publication
OpenAlex ID: W1976956568
DOI: 10.1016/J.JALGEBRA.2012.10.012
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review text: This paper deals with the prime graph \(\Pi(G)\) associated to the conjugacy classes of a finite group \(G\). Recall that the vertices of \(\Pi(G)\) are the prime numbers dividing some conjugacy class size of \(G\) and two distinct primes are joined by an edge when there is some class size divisible by both. The authors study how the set of incomplete vertices (i.e., vertices which are not joined to all the vertices) exerts a strong influence on the structure of the group.NEWLINENEWLINE The main result is a generalization of a former theorem by the same authors [J. Algebra 364, 1-12 (2012; Zbl 1268.20033)], which proved that any finite group such that all the vertices of its prime graph are incomplete must be metabelian. In the paper under review, the authors show that if \(\pi\) denotes the set of incomplete vertices of the prime graph \(\Pi(G)\) of a group \(G\), then \(G\) possesses metabelian Hall \(\pi\)-subgroups, and if \(H\) is such a subgroup, then \(H\cap G''\leq\mathbf F(G)\). The proof relies on the classification of finite simple groups by means of a result due to the first and second author [Isr. J. Math. 174, 403-418 (2009; Zbl 1191.20024)].NEWLINENEWLINE Another theorem is the following. If \(G\) is a finite group, \(p\) is a vertex of \(\Pi(G)\) and \(\pi\) is a set of vertices of \(\Pi(G)\) that are non-adjacent to \(p\) and different from \(p\), then \(G\) is \(\pi\)-solvable, has Abelian Hall \(\pi\)-subgroups and \(\pi\)-length \(1\), and the vertices in \(\pi\) are pairwise adjacent.
description: scientific article; zbMATH DE number 6201455